LMFDB - Embedded newform 2601.2.a.bl.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$20.7690895657$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.4848615424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 16x^{4} - 8x^{2} + 1 $ x^8 - 8x^6 + 16x^4 - 8*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 153)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.437573$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.72291 q^{2} +5.41421 q^{4} -1.47363 q^{5} +3.37849 q^{7} +9.29658 q^{8} +O(q^{10})$	$q+2.72291 q^{2} +5.41421 q^{4} -1.47363 q^{5} +3.37849 q^{7} +9.29658 q^{8} -4.01254 q^{10} -3.55765 q^{11} +1.58579 q^{13} +9.19932 q^{14} +14.4853 q^{16} +3.58579 q^{19} -7.97852 q^{20} -9.68714 q^{22} +1.47363 q^{23} -2.82843 q^{25} +4.31795 q^{26} +18.2919 q^{28} +0.863230 q^{29} +3.24718 q^{31} +20.8489 q^{32} -4.97863 q^{35} +7.07401 q^{37} +9.76376 q^{38} -13.6997 q^{40} -8.58892 q^{41} -6.07107 q^{43} -19.2619 q^{44} +4.01254 q^{46} +7.70154 q^{47} +4.41421 q^{49} -7.70154 q^{50} +8.58579 q^{52} -2.25573 q^{53} +5.24264 q^{55} +31.4084 q^{56} +2.35049 q^{58} -8.16872 q^{59} +1.84776 q^{61} +8.84175 q^{62} +27.7990 q^{64} -2.33686 q^{65} +3.41421 q^{67} -13.5563 q^{70} -10.9258 q^{71} +8.02509 q^{73} +19.2619 q^{74} +19.4142 q^{76} -12.0195 q^{77} -9.68714 q^{79} -21.3459 q^{80} -23.3868 q^{82} -2.25573 q^{83} -16.5309 q^{86} -33.0740 q^{88} +8.82940 q^{89} +5.35757 q^{91} +7.97852 q^{92} +20.9706 q^{94} -5.28411 q^{95} +0.448342 q^{97} +12.0195 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{4}+O(q^{10}) $ 8 * q + 32 * q^4	$ 8 q + 32 q^{4} + 24 q^{13} + 48 q^{16} + 40 q^{19} + 8 q^{43} + 24 q^{49} + 80 q^{52} + 8 q^{55} + 64 q^{64} + 16 q^{67} + 16 q^{70} + 144 q^{76} + 32 q^{94}+O(q^{100}) $ 8 * q + 32 * q^4 + 24 * q^13 + 48 * q^16 + 40 * q^19 + 8 * q^43 + 24 * q^49 + 80 * q^52 + 8 * q^55 + 64 * q^64 + 16 * q^67 + 16 * q^70 + 144 * q^76 + 32 * q^94

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.72291	1.92538	0.962692	−	0.270598i	$-0.0872214\pi$
$2$	2.72291	1.92538	0.962692	+	0.270598i	$0.0872214\pi$
$3$	0	0
$3$	0	0
$4$	5.41421	2.70711
$5$	−1.47363	−0.659025	−0.329513	−	0.944151i	$-0.606884\pi$
$5$	−1.47363	−0.659025	−0.329513	+	0.944151i	$0.606884\pi$
$6$	0	0
$7$	3.37849	1.27695	0.638475	−	0.769642i	$-0.279566\pi$
$7$	3.37849	1.27695	0.638475	+	0.769642i	$0.279566\pi$
$8$	9.29658	3.28684
$9$	0	0
$10$	−4.01254	−1.26888
$11$	−3.55765	−1.07267	−0.536336	−	0.844005i	$-0.680192\pi$
$11$	−3.55765	−1.07267	−0.536336	+	0.844005i	$0.680192\pi$
$12$	0	0
$13$	1.58579	0.439818	0.219909	−	0.975520i	$-0.429424\pi$
$13$	1.58579	0.439818	0.219909	+	0.975520i	$0.429424\pi$
$14$	9.19932	2.45862
$15$	0	0
$16$	14.4853	3.62132
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.58579	0.822636	0.411318	−	0.911492i	$-0.365069\pi$
$19$	3.58579	0.822636	0.411318	+	0.911492i	$0.365069\pi$
$20$	−7.97852	−1.78405
$21$	0	0
$22$	−9.68714	−2.06530
$23$	1.47363	0.307272	0.153636	−	0.988127i	$-0.450902\pi$
$23$	1.47363	0.307272	0.153636	+	0.988127i	$0.450902\pi$
$24$	0	0
$25$	−2.82843	−0.565685
$26$	4.31795	0.846819
$27$	0	0
$28$	18.2919	3.45684
$29$	0.863230	0.160298	0.0801489	−	0.996783i	$-0.474460\pi$
$29$	0.863230	0.160298	0.0801489	+	0.996783i	$0.474460\pi$
$30$	0	0
$31$	3.24718	0.583210	0.291605	−	0.956539i	$-0.405811\pi$
$31$	3.24718	0.583210	0.291605	+	0.956539i	$0.405811\pi$
$32$	20.8489	3.68560
$33$	0	0
$34$	0	0
$35$	−4.97863	−0.841543
$36$	0	0
$37$	7.07401	1.16296	0.581480	−	0.813561i	$-0.302474\pi$
$37$	7.07401	1.16296	0.581480	+	0.813561i	$0.302474\pi$
$38$	9.76376	1.58389
$39$	0	0
$40$	−13.6997	−2.16611
$41$	−8.58892	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$41$	−8.58892	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$42$	0	0
$43$	−6.07107	−0.925829	−0.462915	−	0.886403i	$-0.653196\pi$
$43$	−6.07107	−0.925829	−0.462915	+	0.886403i	$0.653196\pi$
$44$	−19.2619	−2.90383
$45$	0	0
$46$	4.01254	0.591617
$47$	7.70154	1.12338	0.561692	−	0.827346i	$-0.310150\pi$
$47$	7.70154	1.12338	0.561692	+	0.827346i	$0.310150\pi$
$48$	0	0
$49$	4.41421	0.630602
$50$	−7.70154	−1.08916
$51$	0	0
$52$	8.58579	1.19063
$53$	−2.25573	−0.309848	−0.154924	−	0.987926i	$-0.549513\pi$
$53$	−2.25573	−0.309848	−0.154924	+	0.987926i	$0.549513\pi$
$54$	0	0
$55$	5.24264	0.706918
$56$	31.4084	4.19713
$57$	0	0
$58$	2.35049	0.308635
$59$	−8.16872	−1.06348	−0.531738	−	0.846909i	$-0.678461\pi$
$59$	−8.16872	−1.06348	−0.531738	+	0.846909i	$0.678461\pi$
$60$	0	0
$61$	1.84776	0.236581	0.118291	−	0.992979i	$-0.462259\pi$
$61$	1.84776	0.236581	0.118291	+	0.992979i	$0.462259\pi$
$62$	8.84175	1.12290
$63$	0	0
$64$	27.7990	3.47487
$65$	−2.33686	−0.289851
$66$	0	0
$67$	3.41421	0.417113	0.208556	−	0.978010i	$-0.433124\pi$
$67$	3.41421	0.417113	0.208556	+	0.978010i	$0.433124\pi$
$68$	0	0
$69$	0	0
$70$	−13.5563	−1.62029
$71$	−10.9258	−1.29665	−0.648326	−	0.761363i	$-0.724530\pi$
$71$	−10.9258	−1.29665	−0.648326	+	0.761363i	$0.724530\pi$
$72$	0	0
$73$	8.02509	0.939265	0.469633	−	0.882862i	$-0.344386\pi$
$73$	8.02509	0.939265	0.469633	+	0.882862i	$0.344386\pi$
$74$	19.2619	2.23915
$75$	0	0
$76$	19.4142	2.22696
$77$	−12.0195	−1.36975
$78$	0	0
$79$	−9.68714	−1.08989	−0.544944	−	0.838472i	$-0.683449\pi$
$79$	−9.68714	−1.08989	−0.544944	+	0.838472i	$0.683449\pi$
$80$	−21.3459	−2.38654
$81$	0	0
$82$	−23.3868	−2.58264
$83$	−2.25573	−0.247598	−0.123799	−	0.992307i	$-0.539508\pi$
$83$	−2.25573	−0.247598	−0.123799	+	0.992307i	$0.539508\pi$
$84$	0	0
$85$	0	0
$86$	−16.5309	−1.78258
$87$	0	0
$88$	−33.0740	−3.52570
$89$	8.82940	0.935915	0.467957	−	0.883751i	$-0.344990\pi$
$89$	8.82940	0.935915	0.467957	+	0.883751i	$0.344990\pi$
$90$	0	0
$91$	5.35757	0.561626
$92$	7.97852	0.831819
$93$	0	0
$94$	20.9706	2.16295
$95$	−5.28411	−0.542138
$96$	0	0
$97$	0.448342	0.0455222	0.0227611	−	0.999741i	$-0.492754\pi$
$97$	0.448342	0.0455222	0.0227611	+	0.999741i	$0.492754\pi$
$98$	12.0195	1.21415
$99$	0	0
$100$	−15.3137	−1.53137
$101$	−9.29658	−0.925044	−0.462522	−	0.886608i	$-0.653055\pi$
$101$	−9.29658	−0.925044	−0.462522	+	0.886608i	$0.653055\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	14.7424	1.44561
$105$	0	0
$106$	−6.14214	−0.596577
$107$	−5.64167	−0.545401	−0.272700	−	0.962099i	$-0.587917\pi$
$107$	−5.64167	−0.545401	−0.272700	+	0.962099i	$0.587917\pi$
$108$	0	0
$109$	−13.6997	−1.31219	−0.656096	−	0.754678i	$-0.727793\pi$
$109$	−13.6997	−1.31219	−0.656096	+	0.754678i	$0.727793\pi$
$110$	14.2752	1.36109
$111$	0	0
$112$	48.9384	4.62425
$113$	13.6202	1.28128	0.640640	−	0.767841i	$-0.278669\pi$
$113$	13.6202	1.28128	0.640640	+	0.767841i	$0.278669\pi$
$114$	0	0
$115$	−2.17157	−0.202500
$116$	4.67371	0.433943
$117$	0	0
$118$	−22.2426	−2.04760
$119$	0	0
$120$	0	0
$121$	1.65685	0.150623
$122$	5.03127	0.455510
$123$	0	0
$124$	17.5809	1.57881
$125$	11.5362	1.03183
$126$	0	0
$127$	−12.8995	−1.14465	−0.572323	−	0.820029i	$-0.693957\pi$
$127$	−12.8995	−1.14465	−0.572323	+	0.820029i	$0.693957\pi$
$128$	33.9962	3.00487
$129$	0	0
$130$	−6.36304	−0.558075
$131$	20.7355	1.81167	0.905834	−	0.423633i	$-0.139245\pi$
$131$	20.7355	1.81167	0.905834	+	0.423633i	$0.139245\pi$
$132$	0	0
$133$	12.1146	1.05047
$134$	9.29658	0.803102
$135$	0	0
$136$	0	0
$137$	6.57368	0.561627	0.280813	−	0.959762i	$-0.409396\pi$
$137$	6.57368	0.561627	0.280813	+	0.959762i	$0.409396\pi$
$138$	0	0
$139$	−12.7486	−1.08132	−0.540661	−	0.841240i	$-0.681826\pi$
$139$	−12.7486	−1.08132	−0.540661	+	0.841240i	$0.681826\pi$
$140$	−26.9554	−2.27815
$141$	0	0
$142$	−29.7499	−2.49655
$143$	−5.64167	−0.471780
$144$	0	0
$145$	−1.27208	−0.105640
$146$	21.8516	1.80845
$147$	0	0
$148$	38.3002	3.14826
$149$	−19.7210	−1.61561	−0.807805	−	0.589450i	$-0.799344\pi$
$149$	−19.7210	−1.61561	−0.807805	+	0.589450i	$0.799344\pi$
$150$	0	0
$151$	2.58579	0.210428	0.105214	−	0.994450i	$-0.466447\pi$
$151$	2.58579	0.210428	0.105214	+	0.994450i	$0.466447\pi$
$152$	33.3356	2.70387
$153$	0	0
$154$	−32.7279	−2.63729
$155$	−4.78512	−0.384350
$156$	0	0
$157$	−20.7990	−1.65994	−0.829970	−	0.557808i	$-0.811643\pi$
$157$	−20.7990	−1.65994	−0.829970	+	0.557808i	$0.811643\pi$
$158$	−26.3772	−2.09845
$159$	0	0
$160$	−30.7235	−2.42890
$161$	4.97863	0.392371
$162$	0	0
$163$	5.67459	0.444468	0.222234	−	0.974993i	$-0.428665\pi$
$163$	5.67459	0.444468	0.222234	+	0.974993i	$0.428665\pi$
$164$	−46.5022	−3.63122
$165$	0	0
$166$	−6.14214	−0.476722
$167$	17.7882	1.37650	0.688248	−	0.725476i	$-0.258380\pi$
$167$	17.7882	1.37650	0.688248	+	0.725476i	$0.258380\pi$
$168$	0	0
$169$	−10.4853	−0.806560
$170$	0	0
$171$	0	0
$172$	−32.8701	−2.50632
$173$	−24.0403	−1.82775	−0.913875	−	0.405995i	$-0.866925\pi$
$173$	−24.0403	−1.82775	−0.913875	+	0.405995i	$0.866925\pi$
$174$	0	0
$175$	−9.55582	−0.722352
$176$	−51.5335	−3.88449
$177$	0	0
$178$	24.0416	1.80200
$179$	−23.1046	−1.72692	−0.863460	−	0.504417i	$-0.831707\pi$
$179$	−23.1046	−1.72692	−0.863460	+	0.504417i	$0.831707\pi$
$180$	0	0
$181$	5.09494	0.378704	0.189352	−	0.981909i	$-0.439361\pi$
$181$	5.09494	0.378704	0.189352	+	0.981909i	$0.439361\pi$
$182$	14.5882	1.08135
$183$	0	0
$184$	13.6997	1.00995
$185$	−10.4244	−0.766420
$186$	0	0
$187$	0	0
$188$	41.6978	3.04112
$189$	0	0
$190$	−14.3881	−1.04382
$191$	1.78855	0.129415	0.0647075	−	0.997904i	$-0.479389\pi$
$191$	1.78855	0.129415	0.0647075	+	0.997904i	$0.479389\pi$
$192$	0	0
$193$	15.6788	1.12858	0.564291	−	0.825576i	$-0.309150\pi$
$193$	15.6788	1.12858	0.564291	+	0.825576i	$0.309150\pi$
$194$	1.22079	0.0876477
$195$	0	0
$196$	23.8995	1.70711
$197$	6.86246	0.488930	0.244465	−	0.969658i	$-0.421388\pi$
$197$	6.86246	0.488930	0.244465	+	0.969658i	$0.421388\pi$
$198$	0	0
$199$	−25.0489	−1.77567	−0.887834	−	0.460165i	$-0.847790\pi$
$199$	−25.0489	−1.77567	−0.887834	+	0.460165i	$0.847790\pi$
$200$	−26.2947	−1.85932
$201$	0	0
$202$	−25.3137	−1.78107
$203$	2.91642	0.204692
$204$	0	0
$205$	12.6569	0.883993
$206$	2.72291	0.189714
$207$	0	0
$208$	22.9706	1.59272
$209$	−12.7570	−0.882418
$210$	0	0
$211$	0.634051	0.0436498	0.0218249	−	0.999762i	$-0.493052\pi$
$211$	0.634051	0.0436498	0.0218249	+	0.999762i	$0.493052\pi$
$212$	−12.2130	−0.838792
$213$	0	0
$214$	−15.3617	−1.05011
$215$	8.94648	0.610145
$216$	0	0
$217$	10.9706	0.744730
$218$	−37.3029	−2.52647
$219$	0	0
$220$	28.3848	1.91370
$221$	0	0
$222$	0	0
$223$	8.07107	0.540479	0.270239	−	0.962793i	$-0.412897\pi$
$223$	8.07107	0.540479	0.270239	+	0.962793i	$0.412897\pi$
$224$	70.4378	4.70632
$225$	0	0
$226$	37.0865	2.46696
$227$	13.9778	0.927736	0.463868	−	0.885904i	$-0.346461\pi$
$227$	13.9778	0.927736	0.463868	+	0.885904i	$0.346461\pi$
$228$	0	0
$229$	17.4142	1.15076	0.575382	−	0.817885i	$-0.304853\pi$
$229$	17.4142	1.15076	0.575382	+	0.817885i	$0.304853\pi$
$230$	−5.91299	−0.389891
$231$	0	0
$232$	8.02509	0.526873
$233$	19.5147	1.27845	0.639225	−	0.769020i	$-0.279255\pi$
$233$	19.5147	1.27845	0.639225	+	0.769020i	$0.279255\pi$
$234$	0	0
$235$	−11.3492	−0.740339
$236$	−44.2272	−2.87894
$237$	0	0
$238$	0	0
$239$	−14.0817	−0.910870	−0.455435	−	0.890269i	$-0.650516\pi$
$239$	−14.0817	−0.910870	−0.455435	+	0.890269i	$0.650516\pi$
$240$	0	0
$241$	17.7122	1.14094	0.570472	−	0.821317i	$-0.306760\pi$
$241$	17.7122	1.14094	0.570472	+	0.821317i	$0.306760\pi$
$242$	4.51146	0.290007
$243$	0	0
$244$	10.0042	0.640451
$245$	−6.50490	−0.415583
$246$	0	0
$247$	5.68629	0.361810
$248$	30.1876	1.91692
$249$	0	0
$250$	31.4119	1.98666
$251$	19.7210	1.24478	0.622390	−	0.782707i	$-0.286162\pi$
$251$	19.7210	1.24478	0.622390	+	0.782707i	$0.286162\pi$
$252$	0	0
$253$	−5.24264	−0.329602
$254$	−35.1241	−2.20388
$255$	0	0
$256$	36.9706	2.31066
$257$	−13.6145	−0.849251	−0.424625	−	0.905369i	$-0.639594\pi$
$257$	−13.6145	−0.849251	−0.424625	+	0.905369i	$0.639594\pi$
$258$	0	0
$259$	23.8995	1.48504
$260$	−12.6522	−0.784658
$261$	0	0
$262$	56.4608	3.48816
$263$	−18.1260	−1.11770	−0.558848	−	0.829270i	$-0.688756\pi$
$263$	−18.1260	−1.11770	−0.558848	+	0.829270i	$0.688756\pi$
$264$	0	0
$265$	3.32410	0.204198
$266$	32.9868	2.02255
$267$	0	0
$268$	18.4853	1.12917
$269$	−9.80971	−0.598109	−0.299054	−	0.954236i	$-0.596671\pi$
$269$	−9.80971	−0.598109	−0.299054	+	0.954236i	$0.596671\pi$
$270$	0	0
$271$	17.9289	1.08911	0.544553	−	0.838727i	$-0.316699\pi$
$271$	17.9289	1.08911	0.544553	+	0.838727i	$0.316699\pi$
$272$	0	0
$273$	0	0
$274$	17.8995	1.08135
$275$	10.0625	0.606794
$276$	0	0
$277$	−23.2555	−1.39729	−0.698644	−	0.715470i	$-0.746213\pi$
$277$	−23.2555	−1.39729	−0.698644	+	0.715470i	$0.746213\pi$
$278$	−34.7132	−2.08196
$279$	0	0
$280$	−46.2843	−2.76601
$281$	−18.3996	−1.09763	−0.548816	−	0.835943i	$-0.684921\pi$
$281$	−18.3996	−1.09763	−0.548816	+	0.835943i	$0.684921\pi$
$282$	0	0
$283$	−2.93015	−0.174179	−0.0870897	−	0.996200i	$-0.527757\pi$
$283$	−2.93015	−0.174179	−0.0870897	+	0.996200i	$0.527757\pi$
$284$	−59.1545	−3.51017
$285$	0	0
$286$	−15.3617	−0.908358
$287$	−29.0176	−1.71286
$288$	0	0
$289$	0	0
$290$	−3.46375	−0.203398
$291$	0	0
$292$	43.4495	2.54269
$293$	12.6802	0.740784	0.370392	−	0.928876i	$-0.379223\pi$
$293$	12.6802	0.740784	0.370392	+	0.928876i	$0.379223\pi$
$294$	0	0
$295$	12.0376	0.700858
$296$	65.7641	3.82246
$297$	0	0
$298$	−53.6985	−3.11067
$299$	2.33686	0.135144
$300$	0	0
$301$	−20.5111	−1.18224
$302$	7.04085	0.405155
$303$	0	0
$304$	51.9411	2.97903
$305$	−2.72291	−0.155913
$306$	0	0
$307$	15.8995	0.907432	0.453716	−	0.891146i	$-0.350098\pi$
$307$	15.8995	0.907432	0.453716	+	0.891146i	$0.350098\pi$
$308$	−65.0761	−3.70805
$309$	0	0
$310$	−13.0294	−0.740022
$311$	−10.4201	−0.590870	−0.295435	−	0.955363i	$-0.595465\pi$
$311$	−10.4201	−0.590870	−0.295435	+	0.955363i	$0.595465\pi$
$312$	0	0
$313$	−23.4412	−1.32498	−0.662488	−	0.749073i	$-0.730499\pi$
$313$	−23.4412	−1.32498	−0.662488	+	0.749073i	$0.730499\pi$
$314$	−56.6337	−3.19602
$315$	0	0
$316$	−52.4482	−2.95044
$317$	−0.863230	−0.0484838	−0.0242419	−	0.999706i	$-0.507717\pi$
$317$	−0.863230	−0.0484838	−0.0242419	+	0.999706i	$0.507717\pi$
$318$	0	0
$319$	−3.07107	−0.171947
$320$	−40.9653	−2.29003
$321$	0	0
$322$	13.5563	0.755466
$323$	0	0
$324$	0	0
$325$	−4.48528	−0.248799
$326$	15.4514	0.855773
$327$	0	0
$328$	−79.8476	−4.40885
$329$	26.0196	1.43451
$330$	0	0
$331$	−1.14214	−0.0627775	−0.0313887	−	0.999507i	$-0.509993\pi$
$331$	−1.14214	−0.0627775	−0.0313887	+	0.999507i	$0.509993\pi$
$332$	−12.2130	−0.670275
$333$	0	0
$334$	48.4357	2.65028
$335$	−5.03127	−0.274888
$336$	0	0
$337$	17.7122	0.964846	0.482423	−	0.875938i	$-0.339757\pi$
$337$	17.7122	0.964846	0.482423	+	0.875938i	$0.339757\pi$
$338$	−28.5504	−1.55294
$339$	0	0
$340$	0	0
$341$	−11.5523	−0.625593
$342$	0	0
$343$	−8.73606	−0.471703
$344$	−56.4402	−3.04305
$345$	0	0
$346$	−65.4595	−3.51912
$347$	−19.7675	−1.06118	−0.530588	−	0.847630i	$-0.678029\pi$
$347$	−19.7675	−1.06118	−0.530588	+	0.847630i	$0.678029\pi$
$348$	0	0
$349$	−9.72792	−0.520724	−0.260362	−	0.965511i	$-0.583842\pi$
$349$	−9.72792	−0.520724	−0.260362	+	0.965511i	$0.583842\pi$
$350$	−26.0196	−1.39081
$351$	0	0
$352$	−74.1730	−3.95343
$353$	16.5309	0.879853	0.439927	−	0.898034i	$-0.355004\pi$
$353$	16.5309	0.879853	0.439927	+	0.898034i	$0.355004\pi$
$354$	0	0
$355$	16.1005	0.854526
$356$	47.8043	2.53362
$357$	0	0
$358$	−62.9117	−3.32499
$359$	8.82940	0.465998	0.232999	−	0.972477i	$-0.425146\pi$
$359$	8.82940	0.465998	0.232999	+	0.972477i	$0.425146\pi$
$360$	0	0
$361$	−6.14214	−0.323270
$362$	13.8730	0.729150
$363$	0	0
$364$	29.0070	1.52038
$365$	−11.8260	−0.619000
$366$	0	0
$367$	−1.84776	−0.0964522	−0.0482261	−	0.998836i	$-0.515357\pi$
$367$	−1.84776	−0.0964522	−0.0482261	+	0.998836i	$0.515357\pi$
$368$	21.3459	1.11273
$369$	0	0
$370$	−28.3848	−1.47565
$371$	−7.62096	−0.395661
$372$	0	0
$373$	16.0416	0.830604	0.415302	−	0.909684i	$-0.363676\pi$
$373$	16.0416	0.830604	0.415302	+	0.909684i	$0.363676\pi$
$374$	0	0
$375$	0	0
$376$	71.5980	3.69238
$377$	1.36890	0.0705019
$378$	0	0
$379$	−5.80591	−0.298230	−0.149115	−	0.988820i	$-0.547642\pi$
$379$	−5.80591	−0.298230	−0.149115	+	0.988820i	$0.547642\pi$
$380$	−28.6093	−1.46763
$381$	0	0
$382$	4.87006	0.249174
$383$	3.19008	0.163006	0.0815028	−	0.996673i	$-0.474028\pi$
$383$	3.19008	0.163006	0.0815028	+	0.996673i	$0.474028\pi$
$384$	0	0
$385$	17.7122	0.902699
$386$	42.6918	2.17295
$387$	0	0
$388$	2.42742	0.123233
$389$	−2.72291	−0.138057	−0.0690284	−	0.997615i	$-0.521990\pi$
$389$	−2.72291	−0.138057	−0.0690284	+	0.997615i	$0.521990\pi$
$390$	0	0
$391$	0	0
$392$	41.0371	2.07269
$393$	0	0
$394$	18.6858	0.941379
$395$	14.2752	0.718264
$396$	0	0
$397$	2.35049	0.117968	0.0589839	−	0.998259i	$-0.481214\pi$
$397$	2.35049	0.117968	0.0589839	+	0.998259i	$0.481214\pi$
$398$	−68.2057	−3.41884
$399$	0	0
$400$	−40.9706	−2.04853
$401$	−13.9778	−0.698016	−0.349008	−	0.937120i	$-0.613481\pi$
$401$	−13.9778	−0.698016	−0.349008	+	0.937120i	$0.613481\pi$
$402$	0	0
$403$	5.14933	0.256506
$404$	−50.3337	−2.50419
$405$	0	0
$406$	7.94113	0.394111
$407$	−25.1668	−1.24747
$408$	0	0
$409$	36.7990	1.81959	0.909796	−	0.415055i	$-0.136238\pi$
$409$	36.7990	1.81959	0.909796	+	0.415055i	$0.136238\pi$
$410$	34.4634	1.70203
$411$	0	0
$412$	5.41421	0.266739
$413$	−27.5979	−1.35801
$414$	0	0
$415$	3.32410	0.163174
$416$	33.0619	1.62099
$417$	0	0
$418$	−34.7360	−1.69899
$419$	−38.8813	−1.89948	−0.949738	−	0.313047i	$-0.898650\pi$
$419$	−38.8813	−1.89948	−0.949738	+	0.313047i	$0.898650\pi$
$420$	0	0
$421$	10.1716	0.495732	0.247866	−	0.968794i	$-0.420271\pi$
$421$	10.1716	0.495732	0.247866	+	0.968794i	$0.420271\pi$
$422$	1.72646	0.0840428
$423$	0	0
$424$	−20.9706	−1.01842
$425$	0	0
$426$	0	0
$427$	6.24264	0.302103
$428$	−30.5452	−1.47646
$429$	0	0
$430$	24.3604	1.17476
$431$	−19.2619	−0.927811	−0.463906	−	0.885885i	$-0.653552\pi$
$431$	−19.2619	−0.927811	−0.463906	+	0.885885i	$0.653552\pi$
$432$	0	0
$433$	−40.7990	−1.96067	−0.980337	−	0.197330i	$-0.936773\pi$
$433$	−40.7990	−1.96067	−0.980337	+	0.197330i	$0.936773\pi$
$434$	29.8718	1.43389
$435$	0	0
$436$	−74.1730	−3.55224
$437$	5.28411	0.252773
$438$	0	0
$439$	−11.2723	−0.537996	−0.268998	−	0.963141i	$-0.586692\pi$
$439$	−11.2723	−0.537996	−0.268998	+	0.963141i	$0.586692\pi$
$440$	48.7386	2.32352
$441$	0	0
$442$	0	0
$443$	−15.4031	−0.731822	−0.365911	−	0.930650i	$-0.619243\pi$
$443$	−15.4031	−0.731822	−0.365911	+	0.930650i	$0.619243\pi$
$444$	0	0
$445$	−13.0112	−0.616792
$446$	21.9768	1.04063
$447$	0	0
$448$	93.9187	4.43724
$449$	14.2306	0.671583	0.335792	−	0.941936i	$-0.390996\pi$
$449$	14.2306	0.671583	0.335792	+	0.941936i	$0.390996\pi$
$450$	0	0
$451$	30.5563	1.43884
$452$	73.7426	3.46856
$453$	0	0
$454$	38.0601	1.78625
$455$	−7.89505	−0.370126
$456$	0	0
$457$	16.5563	0.774473	0.387237	−	0.921980i	$-0.373430\pi$
$457$	16.5563	0.774473	0.387237	+	0.921980i	$0.373430\pi$
$458$	47.4173	2.21566
$459$	0	0
$460$	−11.7574	−0.548190
$461$	12.6802	0.590575	0.295287	−	0.955409i	$-0.404585\pi$
$461$	12.6802	0.590575	0.295287	+	0.955409i	$0.404585\pi$
$462$	0	0
$463$	36.5269	1.69755	0.848775	−	0.528755i	$-0.177341\pi$
$463$	36.5269	1.69755	0.848775	+	0.528755i	$0.177341\pi$
$464$	12.5041	0.580490
$465$	0	0
$466$	53.1367	2.46151
$467$	19.0603	0.882007	0.441004	−	0.897505i	$-0.354623\pi$
$467$	19.0603	0.882007	0.441004	+	0.897505i	$0.354623\pi$
$468$	0	0
$469$	11.5349	0.532632
$470$	−30.9028	−1.42544
$471$	0	0
$472$	−75.9411	−3.49547
$473$	21.5987	0.993110
$474$	0	0
$475$	−10.1421	−0.465353
$476$	0	0
$477$	0	0
$478$	−38.3431	−1.75377
$479$	−10.6729	−0.487659	−0.243830	−	0.969818i	$-0.578404\pi$
$479$	−10.6729	−0.487659	−0.243830	+	0.969818i	$0.578404\pi$
$480$	0	0
$481$	11.2179	0.511491
$482$	48.2287	2.19676
$483$	0	0
$484$	8.97056	0.407753
$485$	−0.660688	−0.0300003
$486$	0	0
$487$	−25.6285	−1.16134	−0.580670	−	0.814139i	$-0.697209\pi$
$487$	−25.6285	−1.16134	−0.580670	+	0.814139i	$0.697209\pi$
$488$	17.1778	0.777604
$489$	0	0
$490$	−17.7122	−0.800157
$491$	−7.04085	−0.317749	−0.158875	−	0.987299i	$-0.550787\pi$
$491$	−7.04085	−0.317749	−0.158875	+	0.987299i	$0.550787\pi$
$492$	0	0
$493$	0	0
$494$	15.4832	0.696624
$495$	0	0
$496$	47.0363	2.11199
$497$	−36.9127	−1.65576
$498$	0	0
$499$	12.6942	0.568271	0.284135	−	0.958784i	$-0.408293\pi$
$499$	12.6942	0.568271	0.284135	+	0.958784i	$0.408293\pi$
$500$	62.4593	2.79326
$501$	0	0
$502$	53.6985	2.39668
$503$	27.3451	1.21926	0.609629	−	0.792687i	$-0.291318\pi$
$503$	27.3451	1.21926	0.609629	+	0.792687i	$0.291318\pi$
$504$	0	0
$505$	13.6997	0.609628
$506$	−14.2752	−0.634611
$507$	0	0
$508$	−69.8406	−3.09868
$509$	−8.36223	−0.370649	−0.185325	−	0.982677i	$-0.559334\pi$
$509$	−8.36223	−0.370649	−0.185325	+	0.982677i	$0.559334\pi$
$510$	0	0
$511$	27.1127	1.19940
$512$	32.6749	1.44404
$513$	0	0
$514$	−37.0711	−1.63513
$515$	−1.47363	−0.0649357
$516$	0	0
$517$	−27.3994	−1.20502
$518$	65.0761	2.85928
$519$	0	0
$520$	−21.7248	−0.952694
$521$	24.0403	1.05322	0.526612	−	0.850106i	$-0.323462\pi$
$521$	24.0403	1.05322	0.526612	+	0.850106i	$0.323462\pi$
$522$	0	0
$523$	31.5563	1.37986	0.689931	−	0.723875i	$-0.257641\pi$
$523$	31.5563	1.37986	0.689931	+	0.723875i	$0.257641\pi$
$524$	112.266	4.90438
$525$	0	0
$526$	−49.3553	−2.15200
$527$	0	0
$528$	0	0
$529$	−20.8284	−0.905584
$530$	9.05121	0.393159
$531$	0	0
$532$	65.5908	2.84372
$533$	−13.6202	−0.589956
$534$	0	0
$535$	8.31371	0.359433
$536$	31.7405	1.37098
$537$	0	0
$538$	−26.7109	−1.15159
$539$	−15.7042	−0.676428
$540$	0	0
$541$	20.0627	0.862564	0.431282	−	0.902217i	$-0.358061\pi$
$541$	20.0627	0.862564	0.431282	+	0.902217i	$0.358061\pi$
$542$	48.8188	2.09695
$543$	0	0
$544$	0	0
$545$	20.1882	0.864768
$546$	0	0
$547$	−14.6508	−0.626421	−0.313211	−	0.949684i	$-0.601405\pi$
$547$	−14.6508	−0.626421	−0.313211	+	0.949684i	$0.601405\pi$
$548$	35.5913	1.52038
$549$	0	0
$550$	27.3994	1.16831
$551$	3.09536	0.131867
$552$	0	0
$553$	−32.7279	−1.39173
$554$	−63.3225	−2.69032
$555$	0	0
$556$	−69.0237	−2.92726
$557$	31.2733	1.32509	0.662547	−	0.749020i	$-0.269476\pi$
$557$	31.2733	1.32509	0.662547	+	0.749020i	$0.269476\pi$
$558$	0	0
$559$	−9.62742	−0.407196
$560$	−72.1169	−3.04750
$561$	0	0
$562$	−50.1005	−2.11336
$563$	12.4867	0.526250	0.263125	−	0.964762i	$-0.415247\pi$
$563$	12.4867	0.526250	0.263125	+	0.964762i	$0.415247\pi$
$564$	0	0
$565$	−20.0711	−0.844396
$566$	−7.97852	−0.335362
$567$	0	0
$568$	−101.572	−4.26188
$569$	13.8080	0.578863	0.289432	−	0.957199i	$-0.406534\pi$
$569$	13.8080	0.578863	0.289432	+	0.957199i	$0.406534\pi$
$570$	0	0
$571$	39.9397	1.67143	0.835713	−	0.549167i	$-0.185055\pi$
$571$	39.9397	1.67143	0.835713	+	0.549167i	$0.185055\pi$
$572$	−30.5452	−1.27716
$573$	0	0
$574$	−79.0122	−3.29791
$575$	−4.16804	−0.173819
$576$	0	0
$577$	−12.3137	−0.512626	−0.256313	−	0.966594i	$-0.582508\pi$
$577$	−12.3137	−0.512626	−0.256313	+	0.966594i	$0.582508\pi$
$578$	0	0
$579$	0	0
$580$	−6.88730	−0.285980
$581$	−7.62096	−0.316171
$582$	0	0
$583$	8.02509	0.332365
$584$	74.6059	3.08721
$585$	0	0
$586$	34.5269	1.42629
$587$	−3.85077	−0.158938	−0.0794691	−	0.996837i	$-0.525323\pi$
$587$	−3.85077	−0.158938	−0.0794691	+	0.996837i	$0.525323\pi$
$588$	0	0
$589$	11.6437	0.479770
$590$	32.7773	1.34942
$591$	0	0
$592$	102.469	4.21145
$593$	19.2538	0.790661	0.395330	−	0.918539i	$-0.370630\pi$
$593$	19.2538	0.790661	0.395330	+	0.918539i	$0.370630\pi$
$594$	0	0
$595$	0	0
$596$	−106.774	−4.37363
$597$	0	0
$598$	6.36304	0.260204
$599$	26.7619	1.09346	0.546730	−	0.837309i	$-0.315872\pi$
$599$	26.7619	1.09346	0.546730	+	0.837309i	$0.315872\pi$
$600$	0	0
$601$	−23.0698	−0.941036	−0.470518	−	0.882390i	$-0.655933\pi$
$601$	−23.0698	−0.941036	−0.470518	+	0.882390i	$0.655933\pi$
$602$	−55.8497	−2.27626
$603$	0	0
$604$	14.0000	0.569652
$605$	−2.44158	−0.0992645
$606$	0	0
$607$	12.5629	0.509912	0.254956	−	0.966953i	$-0.417939\pi$
$607$	12.5629	0.509912	0.254956	+	0.966953i	$0.417939\pi$
$608$	74.7597	3.03190
$609$	0	0
$610$	−7.41421	−0.300193
$611$	12.2130	0.494085
$612$	0	0
$613$	−26.3137	−1.06280	−0.531400	−	0.847121i	$-0.678334\pi$
$613$	−26.3137	−1.06280	−0.531400	+	0.847121i	$0.678334\pi$
$614$	43.2928	1.74716
$615$	0	0
$616$	−111.740	−4.50214
$617$	43.5550	1.75346	0.876729	−	0.480985i	$-0.159721\pi$
$617$	43.5550	1.75346	0.876729	+	0.480985i	$0.159721\pi$
$618$	0	0
$619$	−9.68714	−0.389359	−0.194679	−	0.980867i	$-0.562367\pi$
$619$	−9.68714	−0.389359	−0.194679	+	0.980867i	$0.562367\pi$
$620$	−25.9077	−1.04048
$621$	0	0
$622$	−28.3730	−1.13765
$623$	29.8301	1.19512
$624$	0	0
$625$	−2.85786	−0.114315
$626$	−63.8282	−2.55109
$627$	0	0
$628$	−112.610	−4.49364
$629$	0	0
$630$	0	0
$631$	17.1421	0.682418	0.341209	−	0.939988i	$-0.389164\pi$
$631$	17.1421	0.682418	0.341209	+	0.939988i	$0.389164\pi$
$632$	−90.0572	−3.58228
$633$	0	0
$634$	−2.35049	−0.0933500
$635$	19.0090	0.754350
$636$	0	0
$637$	7.00000	0.277350
$638$	−8.36223	−0.331064
$639$	0	0
$640$	−50.0977	−1.98029
$641$	−3.55765	−0.140519	−0.0702593	−	0.997529i	$-0.522383\pi$
$641$	−3.55765	−0.140519	−0.0702593	+	0.997529i	$0.522383\pi$
$642$	0	0
$643$	−1.45381	−0.0573327	−0.0286663	−	0.999589i	$-0.509126\pi$
$643$	−1.45381	−0.0573327	−0.0286663	+	0.999589i	$0.509126\pi$
$644$	26.9554	1.06219
$645$	0	0
$646$	0	0
$647$	33.8027	1.32892	0.664461	−	0.747323i	$-0.268661\pi$
$647$	33.8027	1.32892	0.664461	+	0.747323i	$0.268661\pi$
$648$	0	0
$649$	29.0614	1.14076
$650$	−12.2130	−0.479033
$651$	0	0
$652$	30.7235	1.20322
$653$	39.1341	1.53144	0.765718	−	0.643176i	$-0.222384\pi$
$653$	39.1341	1.53144	0.765718	+	0.643176i	$0.222384\pi$
$654$	0	0
$655$	−30.5563	−1.19394
$656$	−124.413	−4.85751
$657$	0	0
$658$	70.8489	2.76198
$659$	−29.2913	−1.14103	−0.570513	−	0.821289i	$-0.693256\pi$
$659$	−29.2913	−1.14103	−0.570513	+	0.821289i	$0.693256\pi$
$660$	0	0
$661$	16.2132	0.630621	0.315310	−	0.948989i	$-0.397891\pi$
$661$	16.2132	0.630621	0.315310	+	0.948989i	$0.397891\pi$
$662$	−3.10993	−0.120871
$663$	0	0
$664$	−20.9706	−0.813816
$665$	−17.8523	−0.692283
$666$	0	0
$667$	1.27208	0.0492551
$668$	96.3093	3.72632
$669$	0	0
$670$	−13.6997	−0.529265
$671$	−6.57368	−0.253774
$672$	0	0
$673$	39.9079	1.53834	0.769168	−	0.639047i	$-0.220671\pi$
$673$	39.9079	1.53834	0.769168	+	0.639047i	$0.220671\pi$
$674$	48.2287	1.85770
$675$	0	0
$676$	−56.7696	−2.18344
$677$	0.967957	0.0372016	0.0186008	−	0.999827i	$-0.494079\pi$
$677$	0.967957	0.0372016	0.0186008	+	0.999827i	$0.494079\pi$
$678$	0	0
$679$	1.51472	0.0581296
$680$	0	0
$681$	0	0
$682$	−31.4558	−1.20451
$683$	−9.09459	−0.347995	−0.173997	−	0.984746i	$-0.555668\pi$
$683$	−9.09459	−0.347995	−0.173997	+	0.984746i	$0.555668\pi$
$684$	0	0
$685$	−9.68714	−0.370126
$686$	−23.7875	−0.908210
$687$	0	0
$688$	−87.9411	−3.35272
$689$	−3.57710	−0.136277
$690$	0	0
$691$	−28.4818	−1.08350	−0.541748	−	0.840541i	$-0.682237\pi$
$691$	−28.4818	−1.08350	−0.541748	+	0.840541i	$0.682237\pi$
$692$	−130.159	−4.94792
$693$	0	0
$694$	−53.8251	−2.04317
$695$	18.7867	0.712619
$696$	0	0
$697$	0	0
$698$	−26.4882	−1.00259
$699$	0	0
$700$	−51.7373	−1.95548
$701$	−8.36223	−0.315837	−0.157918	−	0.987452i	$-0.550478\pi$
$701$	−8.36223	−0.315837	−0.157918	+	0.987452i	$0.550478\pi$
$702$	0	0
$703$	25.3659	0.956693
$704$	−98.8990	−3.72740
$705$	0	0
$706$	45.0122	1.69406
$707$	−31.4084	−1.18124
$708$	0	0
$709$	−24.5236	−0.921003	−0.460502	−	0.887659i	$-0.652330\pi$
$709$	−24.5236	−0.921003	−0.460502	+	0.887659i	$0.652330\pi$
$710$	43.8402	1.64529
$711$	0	0
$712$	82.0833	3.07620
$713$	4.78512	0.179204
$714$	0	0
$715$	8.31371	0.310915
$716$	−125.093	−4.67496
$717$	0	0
$718$	24.0416	0.897226
$719$	18.6515	0.695582	0.347791	−	0.937572i	$-0.386932\pi$
$719$	18.6515	0.695582	0.347791	+	0.937572i	$0.386932\pi$
$720$	0	0
$721$	3.37849	0.125822
$722$	−16.7245	−0.622420
$723$	0	0
$724$	27.5851	1.02519
$725$	−2.44158	−0.0906781
$726$	0	0
$727$	32.1421	1.19209	0.596043	−	0.802953i	$-0.296739\pi$
$727$	32.1421	1.19209	0.596043	+	0.802953i	$0.296739\pi$
$728$	49.8071	1.84597
$729$	0	0
$730$	−32.2010	−1.19181
$731$	0	0
$732$	0	0
$733$	32.8701	1.21408	0.607042	−	0.794670i	$-0.292356\pi$
$733$	32.8701	1.21408	0.607042	+	0.794670i	$0.292356\pi$
$734$	−5.03127	−0.185708
$735$	0	0
$736$	30.7235	1.13248
$737$	−12.1466	−0.447425
$738$	0	0
$739$	12.8995	0.474516	0.237258	−	0.971447i	$-0.423751\pi$
$739$	12.8995	0.474516	0.237258	+	0.971447i	$0.423751\pi$
$740$	−56.4402	−2.07478
$741$	0	0
$742$	−20.7512	−0.761799
$743$	−24.2931	−0.891229	−0.445614	−	0.895225i	$-0.647015\pi$
$743$	−24.2931	−0.891229	−0.445614	+	0.895225i	$0.647015\pi$
$744$	0	0
$745$	29.0614	1.06473
$746$	43.6798	1.59923
$747$	0	0
$748$	0	0
$749$	−19.0603	−0.696450
$750$	0	0
$751$	11.0866	0.404554	0.202277	−	0.979328i	$-0.435166\pi$
$751$	11.0866	0.404554	0.202277	+	0.979328i	$0.435166\pi$
$752$	111.559	4.06814
$753$	0	0
$754$	3.72738	0.135743
$755$	−3.81048	−0.138678
$756$	0	0
$757$	23.9706	0.871225	0.435612	−	0.900134i	$-0.356532\pi$
$757$	23.9706	0.871225	0.435612	+	0.900134i	$0.356532\pi$
$758$	−15.8089	−0.574207
$759$	0	0
$760$	−49.1241	−1.78192
$761$	−36.2520	−1.31413	−0.657066	−	0.753833i	$-0.728203\pi$
$761$	−36.2520	−1.31413	−0.657066	+	0.753833i	$0.728203\pi$
$762$	0	0
$763$	−46.2843	−1.67560
$764$	9.68360	0.350340
$765$	0	0
$766$	8.68629	0.313848
$767$	−12.9538	−0.467736
$768$	0	0
$769$	1.24264	0.0448108	0.0224054	−	0.999749i	$-0.492868\pi$
$769$	1.24264	0.0448108	0.0224054	+	0.999749i	$0.492868\pi$
$770$	48.2287	1.73804
$771$	0	0
$772$	84.8881	3.05519
$773$	5.71948	0.205715	0.102858	−	0.994696i	$-0.467201\pi$
$773$	5.71948	0.205715	0.102858	+	0.994696i	$0.467201\pi$
$774$	0	0
$775$	−9.18440	−0.329913
$776$	4.16804	0.149624
$777$	0	0
$778$	−7.41421	−0.265812
$779$	−30.7980	−1.10345
$780$	0	0
$781$	38.8701	1.39088
$782$	0	0
$783$	0	0
$784$	63.9411	2.28361
$785$	30.6499	1.09394
$786$	0	0
$787$	50.9175	1.81501	0.907506	−	0.420038i	$-0.137983\pi$
$787$	50.9175	1.81501	0.907506	+	0.420038i	$0.137983\pi$
$788$	37.1548	1.32359
$789$	0	0
$790$	38.8701	1.38293
$791$	46.0157	1.63613
$792$	0	0
$793$	2.93015	0.104053
$794$	6.40017	0.227134
$795$	0	0
$796$	−135.620	−4.80692
$797$	42.6321	1.51011	0.755054	−	0.655663i	$-0.227611\pi$
$797$	42.6321	1.51011	0.755054	+	0.655663i	$0.227611\pi$
$798$	0	0
$799$	0	0
$800$	−58.9696	−2.08489
$801$	0	0
$802$	−38.0601	−1.34395
$803$	−28.5504	−1.00752
$804$	0	0
$805$	−7.33664	−0.258583
$806$	14.0211	0.493873
$807$	0	0
$808$	−86.4264	−3.04047
$809$	14.8410	0.521781	0.260891	−	0.965368i	$-0.415984\pi$
$809$	14.8410	0.521781	0.260891	+	0.965368i	$0.415984\pi$
$810$	0	0
$811$	−15.6018	−0.547854	−0.273927	−	0.961750i	$-0.588323\pi$
$811$	−15.6018	−0.547854	−0.273927	+	0.961750i	$0.588323\pi$
$812$	15.7901	0.554124
$813$	0	0
$814$	−68.5269	−2.40187
$815$	−8.36223	−0.292916
$816$	0	0
$817$	−21.7696	−0.761620
$818$	100.200	3.50342
$819$	0	0
$820$	68.5269	2.39306
$821$	23.5346	0.821364	0.410682	−	0.911779i	$-0.365291\pi$
$821$	23.5346	0.821364	0.410682	+	0.911779i	$0.365291\pi$
$822$	0	0
$823$	31.7515	1.10679	0.553393	−	0.832920i	$-0.313333\pi$
$823$	31.7515	1.10679	0.553393	+	0.832920i	$0.313333\pi$
$824$	9.29658	0.323862
$825$	0	0
$826$	−75.1466	−2.61468
$827$	16.5674	0.576107	0.288053	−	0.957614i	$-0.406992\pi$
$827$	16.5674	0.576107	0.288053	+	0.957614i	$0.406992\pi$
$828$	0	0
$829$	3.75736	0.130498	0.0652492	−	0.997869i	$-0.479216\pi$
$829$	3.75736	0.130498	0.0652492	+	0.997869i	$0.479216\pi$
$830$	9.05121	0.314172
$831$	0	0
$832$	44.0833	1.52831
$833$	0	0
$834$	0	0
$835$	−26.2132	−0.907145
$836$	−69.0689	−2.38880
$837$	0	0
$838$	−105.870	−3.65722
$839$	6.50490	0.224574	0.112287	−	0.993676i	$-0.464182\pi$
$839$	6.50490	0.224574	0.112287	+	0.993676i	$0.464182\pi$
$840$	0	0
$841$	−28.2548	−0.974305
$842$	27.6962	0.954475
$843$	0	0
$844$	3.43289	0.118165
$845$	15.4514	0.531544
$846$	0	0
$847$	5.59767	0.192338
$848$	−32.6749	−1.12206
$849$	0	0
$850$	0	0
$851$	10.4244	0.357345
$852$	0	0
$853$	−28.7988	−0.986051	−0.493026	−	0.870015i	$-0.664109\pi$
$853$	−28.7988	−0.986051	−0.493026	+	0.870015i	$0.664109\pi$
$854$	16.9981	0.581664
$855$	0	0
$856$	−52.4482	−1.79264
$857$	−13.8730	−0.473894	−0.236947	−	0.971523i	$-0.576147\pi$
$857$	−13.8730	−0.473894	−0.236947	+	0.971523i	$0.576147\pi$
$858$	0	0
$859$	12.7279	0.434271	0.217136	−	0.976141i	$-0.430329\pi$
$859$	12.7279	0.434271	0.217136	+	0.976141i	$0.430329\pi$
$860$	48.4382	1.65173
$861$	0	0
$862$	−52.4482	−1.78639
$863$	3.65726	0.124495	0.0622473	−	0.998061i	$-0.480173\pi$
$863$	3.65726	0.124495	0.0622473	+	0.998061i	$0.480173\pi$
$864$	0	0
$865$	35.4264	1.20453
$866$	−111.092	−3.77505
$867$	0	0
$868$	59.3970	2.01606
$869$	34.4634	1.16909
$870$	0	0
$871$	5.41421	0.183454
$872$	−127.360	−4.31296
$873$	0	0
$874$	14.3881	0.486686
$875$	38.9749	1.31759
$876$	0	0
$877$	−28.0878	−0.948458	−0.474229	−	0.880402i	$-0.657273\pi$
$877$	−28.0878	−0.948458	−0.474229	+	0.880402i	$0.657273\pi$
$878$	−30.6933	−1.03585
$879$	0	0
$880$	75.9411	2.55997
$881$	8.84175	0.297886	0.148943	−	0.988846i	$-0.452413\pi$
$881$	8.84175	0.297886	0.148943	+	0.988846i	$0.452413\pi$
$882$	0	0
$883$	25.0000	0.841317	0.420658	−	0.907219i	$-0.361799\pi$
$883$	25.0000	0.841317	0.420658	+	0.907219i	$0.361799\pi$
$884$	0	0
$885$	0	0
$886$	−41.9411	−1.40904
$887$	58.0384	1.94874	0.974370	−	0.224953i	$-0.0722227\pi$
$887$	58.0384	1.94874	0.974370	+	0.224953i	$0.0722227\pi$
$888$	0	0
$889$	−43.5809	−1.46165
$890$	−35.4284	−1.18756
$891$	0	0
$892$	43.6985	1.46313
$893$	27.6161	0.924137
$894$	0	0
$895$	34.0476	1.13808
$896$	114.856	3.83707
$897$	0	0
$898$	38.7485	1.29306
$899$	2.80306	0.0934873
$900$	0	0
$901$	0	0
$902$	83.2020	2.77033
$903$	0	0
$904$	126.621	4.21136
$905$	−7.50803	−0.249575
$906$	0	0
$907$	14.3563	0.476692	0.238346	−	0.971180i	$-0.423395\pi$
$907$	14.3563	0.476692	0.238346	+	0.971180i	$0.423395\pi$
$908$	75.6786	2.51148
$909$	0	0
$910$	−21.4975	−0.712634
$911$	−5.99923	−0.198763	−0.0993817	−	0.995049i	$-0.531686\pi$
$911$	−5.99923	−0.198763	−0.0993817	+	0.995049i	$0.531686\pi$
$912$	0	0
$913$	8.02509	0.265592
$914$	45.0814	1.49116
$915$	0	0
$916$	94.2843	3.11524
$917$	70.0547	2.31341
$918$	0	0
$919$	38.1716	1.25916	0.629582	−	0.776934i	$-0.283226\pi$
$919$	38.1716	1.25916	0.629582	+	0.776934i	$0.283226\pi$
$920$	−20.1882	−0.665585
$921$	0	0
$922$	34.5269	1.13708
$923$	−17.3259	−0.570291
$924$	0	0
$925$	−20.0083	−0.657870
$926$	99.4593	3.26844
$927$	0	0
$928$	17.9974	0.590793
$929$	−56.8176	−1.86413	−0.932063	−	0.362296i	$-0.881993\pi$
$929$	−56.8176	−1.86413	−0.932063	+	0.362296i	$0.881993\pi$
$930$	0	0
$931$	15.8284	0.518756
$932$	105.657	3.46090
$933$	0	0
$934$	51.8995	1.69820
$935$	0	0
$936$	0	0
$937$	52.2843	1.70805	0.854026	−	0.520230i	$-0.174154\pi$
$937$	52.2843	1.70805	0.854026	+	0.520230i	$0.174154\pi$
$938$	31.4084	1.02552
$939$	0	0
$940$	−61.4469	−2.00418
$941$	−31.7660	−1.03554	−0.517771	−	0.855519i	$-0.673238\pi$
$941$	−31.7660	−1.03554	−0.517771	+	0.855519i	$0.673238\pi$
$942$	0	0
$943$	−12.6569	−0.412164
$944$	−118.326	−3.85119
$945$	0	0
$946$	58.8113	1.91212
$947$	−4.31615	−0.140256	−0.0701280	−	0.997538i	$-0.522341\pi$
$947$	−4.31615	−0.140256	−0.0701280	+	0.997538i	$0.522341\pi$
$948$	0	0
$949$	12.7261	0.413106
$950$	−27.6161	−0.895984
$951$	0	0
$952$	0	0
$953$	−8.36223	−0.270879	−0.135440	−	0.990786i	$-0.543245\pi$
$953$	−8.36223	−0.270879	−0.135440	+	0.990786i	$0.543245\pi$
$954$	0	0
$955$	−2.63566	−0.0852878
$956$	−76.2413	−2.46582
$957$	0	0
$958$	−29.0614	−0.938932
$959$	22.2091	0.717170
$960$	0	0
$961$	−20.4558	−0.659866
$962$	30.5452	0.984817
$963$	0	0
$964$	95.8978	3.08866
$965$	−23.1046	−0.743764
$966$	0	0
$967$	31.3848	1.00927	0.504633	−	0.863334i	$-0.331628\pi$
$967$	31.3848	1.00927	0.504633	+	0.863334i	$0.331628\pi$
$968$	15.4031	0.495074
$969$	0	0
$970$	−1.79899	−0.0577621
$971$	−9.68360	−0.310762	−0.155381	−	0.987855i	$-0.549660\pi$
$971$	−9.68360	−0.310762	−0.155381	+	0.987855i	$0.549660\pi$
$972$	0	0
$973$	−43.0711	−1.38080
$974$	−69.7840	−2.23603
$975$	0	0
$976$	26.7653	0.856737
$977$	−6.30001	−0.201555	−0.100778	−	0.994909i	$-0.532133\pi$
$977$	−6.30001	−0.201555	−0.100778	+	0.994909i	$0.532133\pi$
$978$	0	0
$979$	−31.4119	−1.00393
$980$	−35.2189	−1.12503
$981$	0	0
$982$	−19.1716	−0.611789
$983$	22.1044	0.705020	0.352510	−	0.935808i	$-0.385328\pi$
$983$	22.1044	0.705020	0.352510	+	0.935808i	$0.385328\pi$
$984$	0	0
$985$	−10.1127	−0.322217
$986$	0	0
$987$	0	0
$988$	30.7868	0.979458
$989$	−8.94648	−0.284482
$990$	0	0
$991$	6.64820	0.211187	0.105594	−	0.994409i	$-0.466326\pi$
$991$	6.64820	0.211187	0.105594	+	0.994409i	$0.466326\pi$
$992$	67.7000	2.14948
$993$	0	0
$994$	−100.510	−3.18797
$995$	36.9127	1.17021
$996$	0	0
$997$	46.3797	1.46886	0.734430	−	0.678685i	$-0.237450\pi$
$997$	46.3797	1.46886	0.734430	+	0.678685i	$0.237450\pi$
$998$	34.5651	1.09414
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bl.1.7		8
3.2	odd	2	inner	2601.2.a.bl.1.2		8
17.11	odd	16		153.2.l.d.19.2	yes	8
17.14	odd	16		153.2.l.d.145.2	yes	8
17.16	even	2	inner	2601.2.a.bl.1.8		8
51.11	even	16		153.2.l.d.19.1	✓	8
51.14	even	16		153.2.l.d.145.1	yes	8
51.50	odd	2	inner	2601.2.a.bl.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.2.l.d.19.1	✓	8	51.11	even	16
153.2.l.d.19.2	yes	8	17.11	odd	16
153.2.l.d.145.1	yes	8	51.14	even	16
153.2.l.d.145.2	yes	8	17.14	odd	16
2601.2.a.bl.1.1		8	51.50	odd	2	inner
2601.2.a.bl.1.2		8	3.2	odd	2	inner
2601.2.a.bl.1.7		8	1.1	even	1	trivial
2601.2.a.bl.1.8		8	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q+2.72291 q^{2} +5.41421 q^{4} -1.47363 q^{5} +3.37849 q^{7} +9.29658 q^{8} +O(q^{10})\)	\(q+2.72291 q^{2} +5.41421 q^{4} -1.47363 q^{5} +3.37849 q^{7} +9.29658 q^{8} -4.01254 q^{10} -3.55765 q^{11} +1.58579 q^{13} +9.19932 q^{14} +14.4853 q^{16} +3.58579 q^{19} -7.97852 q^{20} -9.68714 q^{22} +1.47363 q^{23} -2.82843 q^{25} +4.31795 q^{26} +18.2919 q^{28} +0.863230 q^{29} +3.24718 q^{31} +20.8489 q^{32} -4.97863 q^{35} +7.07401 q^{37} +9.76376 q^{38} -13.6997 q^{40} -8.58892 q^{41} -6.07107 q^{43} -19.2619 q^{44} +4.01254 q^{46} +7.70154 q^{47} +4.41421 q^{49} -7.70154 q^{50} +8.58579 q^{52} -2.25573 q^{53} +5.24264 q^{55} +31.4084 q^{56} +2.35049 q^{58} -8.16872 q^{59} +1.84776 q^{61} +8.84175 q^{62} +27.7990 q^{64} -2.33686 q^{65} +3.41421 q^{67} -13.5563 q^{70} -10.9258 q^{71} +8.02509 q^{73} +19.2619 q^{74} +19.4142 q^{76} -12.0195 q^{77} -9.68714 q^{79} -21.3459 q^{80} -23.3868 q^{82} -2.25573 q^{83} -16.5309 q^{86} -33.0740 q^{88} +8.82940 q^{89} +5.35757 q^{91} +7.97852 q^{92} +20.9706 q^{94} -5.28411 q^{95} +0.448342 q^{97} +12.0195 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{4}+O(q^{10}) \) 8 * q + 32 * q^4	\( 8 q + 32 q^{4} + 24 q^{13} + 48 q^{16} + 40 q^{19} + 8 q^{43} + 24 q^{49} + 80 q^{52} + 8 q^{55} + 64 q^{64} + 16 q^{67} + 16 q^{70} + 144 q^{76} + 32 q^{94}+O(q^{100}) \) 8 * q + 32 * q^4 + 24 * q^13 + 48 * q^16 + 40 * q^19 + 8 * q^43 + 24 * q^49 + 80 * q^52 + 8 * q^55 + 64 * q^64 + 16 * q^67 + 16 * q^70 + 144 * q^76 + 32 * q^94

Introduction

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